Remarks of global wellposedness of liquid crystal flows and heat flows of harmonic maps in two dimensions

Abstract

We consider the Cauchy problem to the two-dimensional incompressible liquid crystal equation and the heat flows of harmonic maps equation. Under a natural geometric angle condition, we give a new proof of the global well-posedness of smooth solutions for a class of large initial data in energy space. This result was originally obtained by Ding-Lin in \cite{DingLin} and Lin-Lin-Wang in \cite{LinLinWang}. Our main technical tool is a rigidity theorem which gives the coercivity of the harmonic energy under certain angle condition. Our proof is based on a frequency localization argument combined with the concentration-compactness approach which can be of independent interest.